## Solving the Equation: (x+1)(x-2)-(4x-3)(x+5)=x(x-9)

This article will guide you through the process of solving the equation **(x+1)(x-2)-(4x-3)(x+5)=x(x-9)**.

### Step 1: Expand the products

We begin by expanding the products on both sides of the equation:

**(x+1)(x-2)**= x² - x - 2**(4x-3)(x+5)**= 4x² + 17x - 15**x(x-9)**= x² - 9x

Now, the equation becomes:
**(x² - x - 2) - (4x² + 17x - 15) = x² - 9x**

### Step 2: Simplify the equation

Next, we simplify the equation by removing the parentheses and combining like terms:

- x² - x - 2 - 4x² - 17x + 15 = x² - 9x
**-3x² - 18x + 13 = x² - 9x**

### Step 3: Move all terms to one side

To solve for *x*, we need to move all terms to one side of the equation:

**-3x² - 18x + 13 - x² + 9x = 0****-4x² - 9x + 13 = 0**

### Step 4: Solve the quadratic equation

We now have a quadratic equation in the form of **ax² + bx + c = 0**. To solve this, we can use the quadratic formula:

**x = (-b ± √(b² - 4ac)) / 2a**

In our equation, a = -4, b = -9, and c = 13. Substituting these values into the quadratic formula, we get:

**x = (9 ± √((-9)² - 4 * -4 * 13)) / (2 * -4)**

**x = (9 ± √(81 + 208)) / -8**

**x = (9 ± √289) / -8**

**x = (9 ± 17) / -8**

This gives us two possible solutions:

**x = (9 + 17) / -8 = -3.25****x = (9 - 17) / -8 = 1**

### Conclusion

Therefore, the solutions to the equation **(x+1)(x-2)-(4x-3)(x+5)=x(x-9)** are **x = -3.25** and **x = 1**.